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I think the empty set satisfies all of the axioms of a vector space except the one about the existence of an additive identity. Is this right?

user209799
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2 Answers2

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The empty set is empty (no elements), hence it fails to have the zero vector as an element.

Since it fails to contain zero vector, it cannot be a vector space.

amWhy
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No! If $(E,+,\cdot)$ is a vector space then $(E,+)$ is an abelian group so it contains a neutral element which is the zero vector hence $E\ne\varnothing$.

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    That's exactly what I said, that the empty set has no zero vector! – user209799 Jan 22 '15 at 20:11
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    It's obvious that the empty set doesn't contain a zero vector since it doesn't contain any element but my answer explains why the empty set isn't a vector space. –  Jan 22 '15 at 20:16
  • That's also what I said in the question, that since the empty set doesn't contain a 0 vector then it isn't a vector space; Thanks anyway, I like your answer since it's technical. – user209799 Jan 22 '15 at 20:17
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    @user209799 Perhaps people are confused because you did not actually write "and therefore it is not a vector space" at the end of your first sentence, although evidently that's what you were thinking. Sometimes it's worth stating the obvious. – David K Jan 22 '15 at 20:19